Set the following:

A = number of apples Sally brought home

P = number of plums Sally brought home

Sally returned home with two times the number of plums than apples.

So: P = 2A

Cost of each apple = 50 cents = $0.5

Cost of each plum = 35 cents = $0.35

Total cost = $12.5

Note that the total cost included the cost of one apple Sally ate on her way home.

The number of apples Sally bought at the market is A+1.

The number of plums Sally bought at the market is P.

(Cost of each apple) * (Number of apples bought) + (Cost of each plum) *(Number of apples bought) = Total Cost = $12.5

0.5*(A+1) + 0.35*P = 12.5 (The $ sign cancelled out on both sides of the equation)

Substitute 2A for P:

0.5*(A+1) + 0.35*(2A) = 12.5

0.5*A + 0.5 + 0.7*A = 12.5

1.2A = 12.5 - 0.5

A = 12/1.2 = 10 (The number of apples Sally brought home)

The number of apples Sally bought at the market = A + 1 = 10 + 1 = 11

P = 2A = 20 (The Number of plums Sally brought home)

The number of plums Sally bought at the market = P = 2A = 20

Verify solution:

$0.5*(11) + $0.35*(20) = $5.5 + $7 = $12.5

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There is more than one way to cut the goose -

Here is another (clearer) way to arrive at the same solution:

A = The number of apples Sally bought at the market

Cost of each apple is 50 cents = $0.5

P = The number of plums Sally bought at the market

Cost of each plum is 35 cents = $0.35

Sally ate one apple (that she bought) on the way home.

So the number of apples Sally brought home is (A-1).

The number of plums Sally brought home is still P.

Sally brought home twice the number of plums than apples.

So the ratio of apples to plums is P = 2*(A-1)

Total Cost = $0.5*A + $0.35*P = $12.5

After removing the $ sign from both sides of the equation we get

0.5*A + 0.35*P = 12.5

Or:

0.5A + 0.35P = 12.5

Substitute 2(A-1) for P gives

0.5A + 0.35*2(A-1) = 12.5

0.5A + 0.7(A-1) = 12.5

0.5A + 0.7A - 0.7 = 12.5

0.5A + 0.7A = 12.5 + 0.7

1.2A = 13.2

A = 13.2/1.2 = 11

P = 2(A-1) = 2(11-1) = 2(10) =20

So:

The number of apples Sally bought at the market = 11

The number of plums Sally bought at the market = 20

Verify solution:

$0.5*(11) + $0.35*(20) = $5.5 + $7 = $12.5